docs(notes): add linear algebra introduction

Author: lzwjava

notes/2025-03-24-linear-algebra-introduction-en.md

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+---
+title: "Linear Algebra - Introduction to Scalars, Vectors, and Matrices"
+lang: en
+layout: post
+audio: false
+translated: false
+generated: true
+---
+
+## **1. Introduction**  
+Linear algebra is a branch of mathematics that deals with **scalars, vectors, matrices, and linear transformations**. It is a foundational subject in various fields, including **science, engineering, computer science, physics, and economics**.  
+
+### **Definition and Importance**  
+Linear algebra is the study of linear equations, vector spaces, and linear transformations. It provides tools to model real-world problems and solve them using systematic methods. Some important applications include:  
+- **Engineering**: Structural analysis, electrical circuit design, control systems  
+- **Physics**: Quantum mechanics, relativity, optics  
+- **Computer Science**: Machine learning, graphics, data compression  
+- **Economics**: Input-output models, optimization problems  
+
+---
+
+## **2. Scalars, Vectors, and Matrices**  
+
+### **Scalars**  
+A **scalar** is a single numerical value, typically representing magnitude. Scalars are used in algebra and calculus, such as:  
+\[
+a = 5, \quad b = -3, \quad c = 2.7
+\]  
+Scalars follow the usual arithmetic rules (addition, multiplication, etc.).
+
+---
+
+### **Vectors**  
+A **vector** is an ordered list of numbers, which can be visualized as an arrow in space. Vectors are used to represent quantities with both **magnitude** and **direction**, such as force, velocity, and acceleration.  
+
+#### **Notation:**  
+A vector in **2D space**:  
+\[
+\mathbf{v} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}
+\]  
+A vector in **3D space**:  
+\[
+\mathbf{u} = \begin{bmatrix} 1 \\ -2 \\ 5 \end{bmatrix}
+\]  
+Vectors can be **added, subtracted, and multiplied by scalars**.
+
+#### **Vector Operations:**  
+1. **Addition and Subtraction**  
+   \[
+   \mathbf{v} + \mathbf{u} = \begin{bmatrix} 3 \\ 4 \end{bmatrix} + \begin{bmatrix} 1 \\ -2 \end{bmatrix} = \begin{bmatrix} 4 \\ 2 \end{bmatrix}
+   \]  
+2. **Scalar Multiplication**  
+   \[
+   2 \mathbf{v} = 2 \begin{bmatrix} 3 \\ 4 \end{bmatrix} = \begin{bmatrix} 6 \\ 8 \end{bmatrix}
+   \]  
+
+---
+
+### **Matrices**  
+A **matrix** is a rectangular array of numbers arranged in rows and columns. Matrices are fundamental in solving systems of equations, computer graphics, and machine learning.
+
+#### **Example of a Matrix:**  
+A **2×3 matrix** (2 rows, 3 columns):  
+\[
+A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}
+\]
+
+#### **Basic Matrix Operations:**  
+1. **Matrix Addition**  
+   \[
+   A + B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}
+   \]
+2. **Scalar Multiplication**  
+   \[
+   3A = 3 \times \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 3 & 6 \\ 9 & 12 \end{bmatrix}
+   \]
+
+---
+
+## **3. Applications of Linear Algebra**  
+
+### **1. Science & Engineering**  
+- **Physics**: Motion equations, electromagnetism, quantum mechanics  
+- **Engineering**: Control systems, robotics, structural analysis  
+
+### **2. Computer Science**  
+- **Machine Learning**: Neural networks, data transformation  
+- **Graphics**: Image processing, 3D modeling  
+
+### **3. Economics**  
+- **Optimization**: Resource allocation, market models  
+- **Statistics**: Regression models, data analysis  
+
+---
+
+## **Conclusion**  
+Linear algebra is a powerful mathematical tool used across various fields. Understanding **scalars, vectors, and matrices** helps in solving real-world problems efficiently. The next step is to explore **determinants, eigenvalues, and linear transformations** for deeper applications.  
+
+Would you like a problem set or further explanations on any topic? 🚀